R- and C-supercyclicity for some classes of operators

The concept of C-supercyclic operators was introduced by Hilden and Wallen in {hilden} and, since then, a multitude of variants have been studied. In {herzog1992linear}, Herzog proved that every real or complex, separable, infinite dimensional Banach space supports a C-supercyclic operator. In the present paper we investigate different variants of supercyclicity, precisely R⁺-, R- and C-supercyclicity in the context of composition operators. We characterize R-supercyclic composition operators on Lᵖ, 1 ≤ p < ∞. Then, we turn our attention to dissipative composition operators, and we show that R- and C-supercyclicity are equivalent notions in this setting and they have a ``shift-like'' characterization.